instruction_seed stringlengths 54 2.22k | source stringclasses 1
value | embeddings sequencelengths 1.02k 1.02k | mean_positive_score float64 0.77 0.86 | mean_negative_score float64 0.76 0.85 | difference_score float64 0 0.04 |
|---|---|---|---|---|---|
Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP > CP$. Let $O_{1}$ and $O_{2}$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. Given that $AB = 8$ and $\angle O_{1}PO_{2} = 150^{\circ}$, then $AP = \sqrt{a} + \sqrt{b}$, where $a$ and $b$ are positive integers. Find $a + b$. | tiger_math | [
0.012361614964902401,
0.0235295370221138,
-0.04475835710763931,
-0.045708026736974716,
0.025984086096286774,
-0.0011414564214646816,
0.00038457888877019286,
0.07461794465780258,
0.055658288300037384,
0.0025481022894382477,
0.0066033536568284035,
0.029118001461029053,
-0.03802909702062607,
... | 0.841065 | 0.817644 | 0.023421 |
Find the values of $c_3$ and $c_4$ such that the following equation holds true:
$$c_3 \begin{pmatrix} 4 \\ -1 \end{pmatrix} + c_4 \begin{pmatrix} -3 \\ 2 \end{pmatrix} = \begin{pmatrix} 1 \\ 7 \end{pmatrix}$$ | tiger_math | [
0.033278629183769226,
0.04414950683712959,
-0.05576392635703087,
-0.0435636043548584,
0.0651063621044159,
-0.025256279855966568,
-0.006799254100769758,
0.06567893177270889,
0.039711203426122665,
-0.009963024407625198,
-0.0016402134206146002,
0.027328358963131905,
-0.031011631712317467,
-0.... | 0.844006 | 0.836264 | 0.007742 |
Let $u$ and $v$ be positive integers satisfying $0 < v < u$. Let $A = (u,v)$, let $B$ be the reflection of $A$ across the line $y = x$, let $C$ be the reflection of $B$ across the y-axis, let $D$ be the reflection of $C$ across the x-axis, and let $E$ be the reflection of $D$ across the y-axis. The area of pentagon $AB... | tiger_math | [
0.014221809804439545,
0.035013724118471146,
-0.05842497944831848,
-0.04196930676698685,
0.036423683166503906,
-0.029116163030266762,
-0.0094853974878788,
0.05179208889603615,
0.06142018362879753,
-0.01032746210694313,
0.020892782136797905,
0.03875325620174408,
-0.031286146491765976,
-0.025... | 0.850359 | 0.829023 | 0.021336 |
Quadrilateral $ABCD$ is a square, and segment $AE$ is perpendicular to segment $ED$. If $AE = 8$ units and $DE = 6$ units, what is the area of pentagon $AEDCB$, in square units? [asy]
size(150);
pair A, B, C, D, E;
A=(0,10);
B=(0,0);
C=(10,0);
D=(10,10);
E=(6.4,5.2);
draw(A--B--C--D--A);
draw(A--E--D);
label("A", A, NW... | tiger_math | [
0.025799209251999855,
0.010997449047863483,
-0.05336267501115799,
-0.05478238686919212,
0.029156457632780075,
-0.02372375875711441,
-0.014731809496879578,
0.060450319200754166,
0.04063091427087784,
0.004238244146108627,
0.01061604917049408,
0.027559053152799606,
-0.035578303039073944,
-0.0... | 0.833538 | 0.821838 | 0.0117 |
Let $\mathbf{v}$ be a vector such that
\[\left\| \mathbf{v} - \begin{pmatrix} 2 \\ 1 \end{pmatrix} \right\| = 5.\]Find the smallest possible value of $\|\mathbf{v}\|.$ | tiger_math | [
0.020631197839975357,
0.03801928833127022,
-0.044878553599119186,
-0.03629925101995468,
0.04911188781261444,
0.000592254102230072,
-0.02757558599114418,
0.0757082924246788,
0.039615534245967865,
-0.011186517775058746,
0.004587564617395401,
0.010368282906711102,
-0.022594662383198738,
-0.00... | 0.833593 | 0.826967 | 0.006626 |
A rectangular playground measures 30 meters in length and 20 meters in width. On a sunny day, Sarah decides to play hopscotch by drawing a straight path diagonal across the playground. She starts at one corner and aims to hop diagonally to the opposite corner. Calculate the square of the length of Sarah's hopscotch pat... | tiger_math | [
0.017362475395202637,
0.023970596492290497,
-0.061627347022295,
-0.0550769604742527,
0.02611885592341423,
-0.027116650715470314,
-0.03185367211699486,
0.04792698845267296,
0.04222200810909271,
-0.03730910271406174,
0.01698482036590576,
0.046365633606910706,
-0.03194998577237129,
-0.0507552... | 0.813158 | 0.809389 | 0.00377 |
A farmer is planning to plant trees in his large, rectangular field. The field measures 120 meters by 180 meters. He intends to plant one tree every 10 meters along the length and every 12 meters along the width. If each tree requires exactly 4 square meters of space around it and no two trees can overlap this space re... | tiger_math | [
0.010303223505616188,
0.027272943407297134,
-0.04850083217024803,
-0.046401601284742355,
0.022481421008706093,
-0.02632066234946251,
-0.0062852539122104645,
0.0644894689321518,
0.027761563658714294,
-0.00796873401850462,
-0.00581571739166975,
0.04312174767255783,
-0.038189612329006195,
-0.... | 0.814245 | 0.808082 | 0.006162 |
In a triangle with sides of lengths $a$, $b$, and $c$, the following equation holds:
$$(a + b + c)(b + c - a) = 3bc.$$
Find the angle opposite the side of length $a$, in degrees. | tiger_math | [
0.026252422481775284,
0.010318612679839134,
-0.03728745877742767,
-0.03928253427147865,
0.021387679502367973,
-0.028902249410748482,
-0.028215983882546425,
0.056015755981206894,
0.03837508335709572,
-0.008860288187861443,
0.02299327775835991,
0.04676975682377815,
-0.0417405441403389,
-0.02... | 0.826486 | 0.8221 | 0.004386 |
"Donna has $n$ boxes of doughnuts. Each box contains $12$ doughnuts.\n\nAfter eating two doughnuts, (...TRUNCATED) | tiger_math | [0.012037977576255798,0.025535468012094498,-0.016737094148993492,-0.033200424164533615,0.01433279272(...TRUNCATED) | 0.848745 | 0.838636 | 0.010109 |
"In the land of Mathemagica, three wizards share a garden where they grow magical sunflowers, moonli(...TRUNCATED) | tiger_math | [0.025106506422162056,0.03624895215034485,-0.03941749781370163,-0.07159307599067688,0.03726276010274(...TRUNCATED) | 0.819847 | 0.814203 | 0.005643 |
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